But one of the questions in my example sheet said think of an example of a disconnected subspace T of a topological space S for which there are no nonempty subsets A,B of T such that [itex]A\cup B = T[/itex] but [itex]\bar{A} \cap \bar{B} = [/itex]∅

Surely if [itex] \bar{A} = A = A^o [/itex] then [itex] \bar{A} \cap \bar{B} = A \cap B = ∅ [/itex] which is the definition of disconnected so there are no examples?

It helps if you have the actual example that you're going to work with. As you've observed this phenomenon is atypical behavior of disconnected spaces, so you really need to exploit the fact that you're working in a larger topology.

Think a bit about the example I gave with A... can you think of a disconnected subset T in R such that when you take the closure of its two separated halves, they intersect (hint: this is the same as just taking the closure of T)